i
The performance of Bayesian VAR Markov switching and
logistic regression models with Monte Carlo simulated data
By
Katleho Makatjane
(23085738)
A dissertation submitted in fulfilment for the Degree Master of
Commerce in Statistics to the Faculty of Commerce and
Administration, School of Economic and Decision Sciences of
the North-West University (Mafikeng Campus)
Supervisor
: Prof N D Moroke
Co-supervisor: Mr L.D Xaba
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Declaration
I, Katleho Makatjane, hereby declare that this dissertation is the result of my own investigation except where otherwise stated. I also declare that it has never been submitted previously as a whole or in part for any other degree at North West University or any other institution.
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Acknowledgements
Undertaking this dissertation has been an inspiring and heartwarming task. Special thanks are due to my supervisors Professor N.D Moroke and Mr L.D Xaba. You have provided a meaningful guidance and support with your constant and unconditional patience and encouragement. To My fiancé, Hopolang Lenoesa, you never left my side and you have always given me that support that one ever needs. Even when it was difficult to go through my study, you always gave me encouraging words that I will be all right. Special thanks are due to the Lord all mighty, for giving me strength and power to do all this wonderful work and being with me for the entire master’s program. A big thank you to my family, especially Professor T.J Makatjane and Mrs Makatjane, and all my friends. Thank you all.
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Dedication
This piece of writing is dedicated to my family, and friends. A special feeling of gratitude goes to my loving parents Tiisetso and Makatleho Makatjane, my fiancé Hopolang Lenoesa, my nephew Matiisetso Makatjane. A special dedication goes to Professor N.D Moroke and Mr LD Xaba for giving all they have for me to attain my statistical skills. I am who I am today because of your unconditional and constant help throughout my whole career.
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Table of Contents
Declaration... ii
Acknowledgements ... iii
Dedication ... iv
List of Tables ... viii
Acronyms and abbreviations ... ix
Abstract ... xi
CHAPTER 1 ... 1
ORIENTATION OF THE STUDY ... 1
1.1 Introduction ... 1
1.2 Problem statement ... 2
1.3 The study aims and objectives ... 3
1.4 Significance of the study ... 3
1.5 Organization of the study ... 4
CHAPTER 2 ... 5
EMPIRICAL AND THEORETICAL LITERATURE... 5
2.1 Introduction ... 5
2.2 Nonlinear models ... 5
2.3 Monte Carlo Simulation ... 7
2.4 Overview of regime switching models ... 9
2.5 Markov switching model and extensions ... 11
2.5.1 AR enhanced Markov-switching model ... 11
2.5.2 VAR enhanced Markov-Switching model ... 14
2.6 Empirical literature ... 16
2.6.1 Empirical literature on Markov-switching vector autoregressive model ... 16
2.6.2 Empirical literature on Logistic regression model ... 20
2.7 Concluding remarks ... 23
CHAPTER 3 ... 24
METHODOLOGY ... 24
3.1 Introduction ... 24
3.2 Data simulation ... 25
3.2.1 Preliminary data analysis ... 26
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3.2.3 Nonlinear unit root test ... 27
3.2.4 Nonlinearity testing ... 29
3.3 Prior tests ... 30
3.3.1 Information criterion for lag length selection and Structural break point test... 30
3.3.2 The reset test ... 31
3.3.3 CUSUM test ... 33
3.4 Primary data analysis ... 34
3.4.1 Theoretical framework ... 35
3.4.2 The expectation maximum (EM) algorithm ... 38
3.4.3 Logistic regression... 40
3.4.4 Computation of marginal effects ... 41
3.5 Model diagnostics checks... 42
3.5.1 Normality ... 42
3.5.2 Serial correlation ... 43
3.5.3 Heteroscedasticity ... 44
3.6 Forecasting with 𝐌𝐒(𝐪) − 𝐁𝐕𝐀𝐑(𝐩) ... 45
3.6.1 Evaluating the Performance of MS(q)-BVAR(p) ... 46
3.7 Concluding remarks ... 46
CHAPTER 4 ... 48
DATA ANALYSIS AND RESULTS ... 48
4.1 Introduction ... 48
4.2 Preliminary results ... 49
4.2.1 Exploratory data analysis results ... 49
4.2.2 Graphical presentation of data ... 49
4.2.3 Nonlinear and nonlinear unit root tests results ... 51
4.3 Empirical results ... 52
4.3.1 Maximum lag length selection and model estimation ... 53
4.3.2 Estimation of 𝐁𝐀𝐑(𝟏) model and RAMSEY reset test results ... 54
4.3.3 Parameter stability test results of 𝐁𝐀𝐑(𝟏) ... 55
4.3.4 Structural breakpoints test results ... 56
4.3.5 𝐌𝐒(𝟐) − 𝐁𝐕𝐀𝐑(𝟏) model estimation results ... 58
4.3.6 Logistic regression results ... 60
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4.5 Forecasting performance of 𝐌𝐒(𝟐) − 𝐁𝐕𝐀𝐑(𝟏) ... 62
4.6 Assessment performance of the EWS ... 63
4.6.1 Predictive power logistic regression models ... 63
4.7 Concluding remarks ... 65
CHAPTER 5 ... 66
CONCLUSIONS AND RECOMMENDATIONS ... 66
5.1 Introduction ... 66
5.2 Research objectives and conclusions ... 66
5.4 Limitations ... 70
5.5 Summary ... 71
REFERENCES ... 72
Appendix A Forecasts of 𝐌𝐒(𝟐) − 𝐁𝐕𝐀𝐑(𝟏) model ... 83
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List of Tables
Table number Table Description page
Table 3.1 Critical values of KSS-NADF 29
Table 4.1 Exploratory data analysis 50
Table 4.2 KSS-NADF test 51
Table 4.3 Nonlinearity test 52
Table 4.4 Lag-Length selection 53
Table 4.5 Estimated BAR (1) Models 54
Table 4.6 Multiple break point test 56
Table 4.7 Periods of structural breaks 57
Table 4.8 MS-BVAR(1) parameter estimates for inflation 58
Table 4.9 Estimated logit model 60
Table 4.10 Model diagnostic checking 61
Table 4.11 Forecasting performance of MS-BVAR (1) 62
Table 4.12 Forecasts of EWS model 63
Table 4.13 Model power test 64
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Acronyms and abbreviations
AIC Akaike information criterion ANN Artificial neural network ANSR Adjusted noise-to signal ratio
AR Autoregressive
ARCH Autoregressive conditional heteroscedasticity ARMA Autoregressive moving average
BDS Brock-Dechert-Scheinkman
BVAR Bayesian vector autoregressive CDF Cumulative density function
CUSUM Cumulative sum
DGP Data generating process
EM Expectation maximisation
EWS Early warning system
EXAR Exponential autoregressive
GARCH Generalized autoregressive conditional heteroscedasticity GLM Generalized linear model
HMM Hidden Markov model
HQC Hannan-Quinn criterion
IV Input Variable
JB Jarque-Bera
KSS-NADF Kapetanios, Shinn-Shell nonlinear augmented Dickey-fuller
LA Logistic analysis
LRM Logistic regression model
MAE Mean absolute error
MAPE Means absolute percentage error MCMC Markov chain Monte Carlo MDA Multiple discriminant analysis
MLE Maximum likelihood
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MS-BVAR Markov-switching Bayesian vector autoregressive
MSE Mean square error
MS-GARCH Markov-switching generalized autoregressive conditional heteroscedasticity
MSM Markov-switching model
MSR Markov-switching regression
MS-SVAR Markov-switching structural vector autoregressive MS-VAR Markov -switching vector autoregressive
NCD Normal cumulative distribution OLS Ordinary least square
PCCC Percent of Crisis Correctly Called PFA Percent of False Alarm
PNCC Percent of Non-Crisis Correctly Called POCC Percent of Observation Correctly Called
PRGNS Probability of an Event of High Inflation Given No Signal PRGS Probability of an Event of High Inflation Given a Signal RESET Regression specification error test
RMSE Root mean square error RSA Republic of South Africa
SA South Africa
SARB South African reserve bank SBC Schwarz Bayesian criterion STAR Smooth transition autoregressive SVAR Structural vector autoregressive
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Abstract
In this study, the main intention is to build an early warning system (EWS) model for inflation in South Africa using the findings from the Markov-switching Bayesian vector autoregressive (MS-BVAR) on logistic regression model. Monte Carlo experimental methods are used to simulate both the inflation rate and repo rate of South Africa. In total, the procedure simulated 210 observations for the period January 1999 to June 2016. For this data generating process, the study followed a Gibbs sampling technique. Prior to model estimation, preliminary test of nonlinearity called the Brock Dechert Scheinkman (BDS) test was employed and the results confirmed the data to be nonlinear and suitable for MS-BVAR method. The Kapetanios-Shin-Snell nonlinear augmented Dickey-Fuller (KSS-NADF) also confirmed the presence of nonlinear unit root in the simulated series. Moreover, the RESET test, CUSUM and Bai Perron multiple break point tests were also calculated to determine if there is structural change in the data and that the model is correctly specified.
With the attempt to build an early warning system (EWS) model, the study estimates the MS − BVAR(1) model of two regime shifts. This model serves as a primary tool in detecting regime shifts in inflation in terms of low and high regimes. The results of the MS(2) − BVAR(1) indicates that the SA inflation might be in low inflation regime for the period of 11 years and 4 months. Furthermore, the results of the logistic regression revealed that the repo rate is not a good tool to predict inflation rate. The results of the marginal effects of the repo rate towards inflation rate implied that if everything held constant, a 1% increase in repo in a month increases inflation by 81%.
Similar results were also reported by several authors such as Mboweni et al. (2008); Gupta and Komen (2009); and Bonga-Bonga and Kabundi (2015). In predicting the possibility of inflation crisis in SA, the assessment of the EWS model confirmed that only 57% of the inflation crises are correctly called for by the in-sample model compared to the 45% of correctly called by out-of-sample model forecasts.
The study concluded that combating inflation rates in South Africa (SA) using variables such as repo rate might not be a good idea as this might also increase the likelihood of SA being be into inflationary. Finally, the study recommends the enhancement of error correction model to the MS-BVAR model when including other determinants of inflation rate in the analysis. The study might
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provide a clearer picture about both the long-term and short-term relationships between inflation and related variables. Such findings might be used by policy makers to embark on strategies to combat the anticipated inflationary crisis in South Africa.
Key words: Early warning system, Markov-switching Bayesian vector autoregressive, Monte
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CHAPTER 1
ORIENTATION OF THE STUDY
1.1 Introduction
The global monetary authorities have targets for smoothing inflation rates including high prices. South Africa is not an exception in striving to this accomplishment. Volatile inflation rates twist the judgement that consumers make about their living standards, and this is evident from past studies which reported about the importance of prices stability. The slow economic growth rates are subdued to markets failure which results from these distortions. Money circulation in the economy for buying goods and services slowly but surely wear off because of high inflation which in the long run harms the households specifically those with low incomes. Nevertheless, keeping a low inflation rate and stabilising it supports sustainable economic growth.
Since consumer inflation is the main indicator in determining the sustainability of economic growth, all trusts of the rate cresting underneath 10% in the first quarter of 2008 have now been surrendered. Inflation rate of South Africa is being driven by four fundamental powers such as food prices, up 70% comprehensively; oil prices (up 65%); the rand (down 15%) against an extremely feeble dollar so far this year Svensson (2005). Other powers are a solid domestic consumer demand during an era when output is obliged by bottlenecks and prominently power and skilled labour. Going ahead, the stress is that proposed power price treks of about 50-60% in 2008/09 guaranteed that inflation stays in twofold figures, particularly if food price stays high Hyvonen (2004).
To exceptionally combat inflation rate to low rates, Bernanke and Woodford (1997) indicated that policy instruments are changed by significant monetary policy lag. According to Mishkin and Schmidt-Hebbel (2007), the persistent increment of South African’s (SA) repo rate caused the inflation rate to develop outside the 3%-6% interval since early 2007. However, the expectation was that the repo rate will sooth inflation rate. In order to achieve best approaches, Svensson (2010) suggested: (1) algebraic growth targets, (2) significant fiscal strategy that provides part of the expansion forecasts to be practiced; and lastly (3) selection of the short-run enthusiasm rate as the best instrument for fiscal strategy.
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1.2 Problem statement
In 1994 the first democratic government of South Africa inherited a persistently high inflation above ten percent per annum. These high inflation rates caused a regressive effect on lower-income families and older people in society because prices for food and domestic utilities such as water and heating rose at a rapid rate Psaradakis and Sola (1998). However, the year 2000 was marked by the approval of an inflation targeting framework as the anchor of monetary policy in South Africa (Bonga-Bonga and Kabundi, 2015). In collaboration with the South African Reserve Bank (SARB), the minister of finance decided the target which was to achieve the average of 3%-6% interval by the year 2000. Previously the SARB was using the repo rate as the policy instrument to control the level of inflation within the chosen interval. However, the effectiveness of the repo rate as policy instrument to control the inflation rate has not been criticized in SA only, but also universally according to Bonga-Bonga and Kabundi (2015). The reason for repo rate being criticized is that altering the repo rate in the monetary sector, affects short-term liquidity in the monetary system, which quickly has an effect on all other rates.
The SA inflation rate went out of the interval in the early 2007 where it then increased from 3% to 8.6%. The recurrent upturn in the repo rate in order to curtail the inflation rate has speeded up its trend rather than subduing it. This is a period in which SA had encountered high production levels because it experienced far above the ground interest rates, increasing inflation, a prolonged fall in prices and trade of housing and vehicle markets and also a degrading of business and consumer firm trust indicators. These problems could have been avoided if they were detected earlier before inflation rates got out of control. It is clear, given these reasons, that there is still a gap with regards to this sector, especially when it comes to controlling it. Reasons may be studies done around warning systems in inflation are fragmented if at all they do not exist or that the methods used are not as effective as possible. This prompted the undertaking of this study where statistical methods such as logistic regression are explored in conjunction with the Monte Carlo simulated inflation rates and estimates Markov-switching model to develop an early warning system and determine the likelihood of future high inflation rate in South Africa. The findings could be useful in alerting the policy makers well in advance about the anticipated inflation rates and as a result, strategies to curb such problems could be devised beforehand.
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1.3 The study aims and objectives
The overarching aim of this study is to explore MS-BVAR model with two regimes to estimate and forecast the inflation rate regimes within the country and finally use logistic regression model to quantify the possibility of the future occurrence of high inflation.
To help address the problem, the following secondary objectives are formulated:
• To simulate the inflation and repo rates data using Monte Carlo method.
• To estimate a Markov-Switching Bayesian vector autoregressive model using the inflation rate and repo rate of South Africa
• To estimate the logistic regression model as a classifier of inflation crises on the basis of repo rate.
• To formulate suggestions for policy and future studies.
1.4 Significance of the study
The study will contribute to literature of early warning system (EWS) by envisioning the probability of inflation crises in the Republic of South Africa (RSA). The expected duration of high and low inflation will be distinguished together with the transition probabilities of the high and low regimes. This monitoring tool so called EWS comprises a detailed definition of a crisis and a mechanism for causing likelihoods of crises (Edison, 2000). The upgrading of EWS for use in economic policy invention will transmit mainly to the extrapolation of financial crises and economic crises.
The warning signs that are driven by a logistic regression model (LRM) will also help in detecting the inflation turbulent for the following 5-years period. The use of the 5-year period will give a larger sample as the study is focused on the monthly simulated series and this also helps in protecting the normality assumption. Given these reasons, the study contributes to the methodology of EWS and fills the gap in early warning signs of inflation crisis. Both scholars and policy makers will benefit from this study.
From the findings of the study, suggestions will be given towards policies of the monetary policy framework. For the monetary policy committee (MPC), the study will develop the models that will
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help them to enrol into the EWS framework for inflation. To assess the inflation environment outlook risk, the South African reserve bank (SARB) may also use this EWS model to substitute the existing inflation toolkit.
1.5 Organization of the study
This study consists of five chapters which are presented as follows:
Chapter 1: The chapter provides the introduction to the study and it also provides a background
that creates relevance for the study. In this section, the study objectives are outlined, significance of the study together with the limitations to the study are also explained.
Chapter 2: This chapter presents the review of concepts of the literature regarding the MS-BVAR
models used to model the inflation regime episodes and the logistic regression model that will help in predicting the likelihood of occurrence of high and low inflation within the in-sample and out-of-sample forecasts.
Chapter 3: The chapter entails the methodology adopted for this study. Detailed theory of
statistical tests that are used to analyses data in developing the EWS are discussed.
Chapter 4 This chapter gives an outline of empirical analysis and interpretation of results.
Chapter 5: This chapter outlines the discussion of findings and provides the conclusion to the
whole study and recommendations for future studies and policy.
1.6 Conclusion
The chapter introduced the study by providing a motivation for the study and defining the problem. The study objectives were outlined on the basis of the problem. Also listed were the study scope limitations and delimitations and further the significance of the study was explored. The next chapter discusses theoretical underpinnings adopted by the study and empirical literature on the subject.
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CHAPTER 2
EMPIRICAL AND THEORETICAL LITERATURE
2.1 Introduction
The financial chaos that hit developing markets in the latest decades has initiated the need for precise country hazard assessment (Fuertes and Kalotychou, 2007). To explain and predict the crisis of the country including the currency crises, several models have been developed by a number of studies worldwide (Kumar et al., 2003). In observing the crises empirically, it is significant to be indistinct on how a crisis is defined. The purpose of the present study is to predict the possibility of inflation crisis in South Africa. The models used for determining the early warning signals of inflation are discussed in this chapter together with those used for obtaining regime switches of inflation rates.
The remaining part of this chapter is as follows: Section 2.2 gives an overview of the nonlinear time series models. Section 2.3 discuses Monte Carlo simulation. Section 2.4 presents an overview of regime switching models while section 2.5 discuses Markov-switching model and extensions. Section 2.6 discusses the empirical literature and finally section 2.7 presents concluding remarks for the chapter.
2.2 Nonlinear models
According to economic theory, a number of time series variables are expected to be nonlinearly related (Kleiber and Zeileis, 2008). Most business cycles experience a sharper recession than salvages in key macroeconomic variables. The variable of interest for the current study is inflation rate. Standard autoregressive moving average (ARMA) relies on linear difference equations, new dynamic area of time series econometrics seems to be growing.
Jonathan and Kung-Sik (2008) suggested that a linear ARMA model should conform to normal distribution standards. However, this linear, normal process suffers some limitations because the ARMA models are stationarised before estimation, they are therefore characterised by their mean and autocovariance function, hence the process reversed has the same distribution as the original
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process. The latter is known as time reversibility. Conversely, nonlinear time series models generally display rich dynamical structure. Indeed Jonathan and Kung-Sik (2008) showed that a very simple nonlinear deterministic difference equation may admit chaotic solutions in the sense that its time series solutions are sensitive to the initial values which may appear to be distinguishable from a white noise sequence based on correlation analysis.
Nonlinear time series analysts thus may provide more accurate predictions which can be very substantial in certain parts of the state space and shed novel insights on the underlying dynamics of the data. Therefore, the analysis of nonlinear time series was earnestly initiated in the late 1970s in which the need for modelling the nonlinear dynamics shown in the real data was prompted by (Tong, 2012). Makridakis and Hibon (2000) warned that linear models are not suitable to epitomize abundant nonlinear patterns; perhaps, asymmetry, high moment structures, time varying, asymmetric cycles, and jumps or breaks in a time series.
Due to unexpected constraints, some researchers more often than not, change the problem of nonlinearity to suit the linear models for the purpose of their studies (Kantz and Schreiber, 2004). While linear models on the one hand are helpful for much research, nonlinearity swarms our consistent life and ought not to be overlooked on the other hand. The motivation behind this is that, nonlinear models are now being given attention specifically business cycle models such as Markov-Switching models and many more.
Since, there exist a number of nonlinear time series models, it is customary to first test the linear hypothesis against the alternative of nonlinear before any modelling can be done (Kantz and Schreiber, 2004). Two nonlinear models such as Markov-Switching autoregressive (MS-AR) and Logistic Regression model have been considered by Cruz and Mapa (2013) with the aim of developing an early warning system model. The forecasts were combined by the regime switching of inflation and the likelihood of the occurrence of the inflation crisis.
Because Zhang (2003) has indicated the importance of combining linear models together with nonlinear models, by following the combination style of Zhang, the study combines the Bayesian Vector Autoregressive (BVAR) with Markov Switching model (MSM). This approach helps in
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addressing the complexity of serially correlated structures of the data set and dynamic changes in the same given series such as structural changes.
The combination for linear models together with some nonlinear models has gained some popularity in recent methodologies. However, this idea has been implemented decades ago. Terui and Van Dijk (2002) have shown that a combination of a Threshold autoregressive model and exponential autoregressive (ExpAR) have complete each other in their performances, and lastly that the combined forecasts from the models can be based locally on linear or nonlinear model which is far much important for time series that exhibits structural changes.
Aside from the MSM and logistic regression models, there are some nonlinear parametric models.; these include autoregressive Conditional heteroscedasticity (ARCH) and general autoregressive conditional heteroscedasticity (GARCH), artificial neural network (ANN), threshold autoregressive and Smooth transition autoregressive (STAR) which have been used for forecasting for some time. In any case, the greater part of the nonlinear statistical procedures require that the nonlinear model must be determined before the estimation of the parameters is done (Xaba et al., 2015).
According to Asteriou and Hall (2015), this pretesting for nonlinearity can help one to be protected from overfitting the data. Recursive estimation and the CUSUM test are frequently used for detecting nonlinearities. A number of additional procedures that have been developed to determine if the data seem to be nonlinear are used in the current study which are the RESET test and BDS test. In which the reset test was developed by Ramsey (1969) to test for the specification errors in classical linear models. in which the null hypothesis is that the model is correctly specified as a linear model. On the other hand, the BDS test was developed by Brock et al. (2001) to test for the independence based on the correlation dimension.
2.3 Monte Carlo Simulation
Monte Carlo simulations are used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables. These contain the dynamic panel group and time specific effects. Smith et al. (2013) have shown that the data generating process has 4 steps. These steps are the initialisation, importance sampling step, selection step and
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Markov transition step. Since this is a sequential method of Monte Carlo, the initialisation step is to initialize the random variable X to some value and then the variable is picked randomly and resampled.
The ultimate assumption is that all evidence about the data generating process (DGP) contained in the original sample of data is also confined in the distribution of these simulated samples. Resampling from one sample is corresponding to generating completely some new random samples from the population. Another way to think about this is that if the sample of data at hand is a reasonable representation of the population, then the distribution of parameter estimates produced from running a model on a series of resampled data sets will provide a good approximation of the distribution of that statistics in the population (Doucet et al., 2001).
The role of Monte Carlo methods has increased in importance during the past several years. According to Gentle (2013),many advances in the field of random number generation and Monte Carlo methods have been made. These methods assume a focal part in the rapidly developing sub disciplines of the computational physical sciences, the computational life sciences, and the other computational sciences. The developing force of computers and the developing simulation methodology have prompted the acknowledgment of computation as a third approach for propelling the characteristic sciences, together with theory and customary experimentation.
Generation of random numbers is also at the heart of many standard statistical methods. The random sampling required in most analyses is usually done. The computations required in Bayesian analysis have become viable because of Monte Carlo methods. This has led to much wider applications of Bayesian statistics, which, in turn, has led to development of new Monte Carlo methods and to refinement of existing procedures for random number generation (Zio, 2013).
Hammersley (2013) has indicated that Monte Carlo methods represents the solution of a problem as a parameter of the hypothetical population and by using random numbers as a sequence. The sample is drawn from the population from which statistical estimates of the parameter can be obtained. The standards for Monte Carlo experiments in statistics were set by Sawilowsky and Fahoome (2003) and Sawilowsky (2003) with the aim of improving the randomization and permutation test.
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Diebold and Inoue (2001) used Monte Carlo experiments to show that the stochastic regime switching is easily confused with long memory even asymptotically as long as a small amount of regime occurs. For their study, Diebold and Inoue, simulated 10,000 realisation from various stochastic regimes switching. They further characterised the finite sample inference with a standard estimator of long memory parameter. The current study does simulation of data through the Gibbs sampling.
2.4 Overview of regime switching models
These types of models can be usefully divided into two categories. The Threshold models, which are modified by Tong (2012) were firstly introduced by Tong (1978) with the assumption that regime shifts are triggered by the level of observed variables in relation to an unobserved threshold. Markov switching models (MSM) which were introduced to econometric modelling by Goldfeld and Quandt (1973), Cosslett and Lee (1985) and Hamilton (1989), with the assumption that the regime shifts evolve according to a Markov Chain.
Piger (2009) accentuates that there is substantial interest in modelling the dynamic behaviour of macroeconomic and financial quantities that are observed over time. The greatest challenge is that time series data likely undergo changes in their behaviour over reasonably long sampled periods. Because of the potential shifts of economic time series, constant parameter time series models are becoming inadequate for describing their evolution hence there was a need to design parameter variation models.
MSM is a regime-switching model in which the shifts between regimes evolve according to an unobserved Markov Chain(MC). According to Piger (2009), regime switching models are time series models in which parameters are allowed to take different values in each of some fixed number known as regimes. A stochastic process assumed to have generated the regime shifts is included as part of the model, which then allows for model-based forecasts that incorporate the possibility of future regime shift.
Simple regime switching models have never found their statistical significance in both empirical and theoretical research in earlier research except their alternative models such as AR models.
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However, scholars such as Kim and Nelson (2001), Kim et al. (2005) and Hamilton (2005) had found some strong statistical significance for both empirical and theoretical research. With that note, Hamilton (2005) focused on the alternative measures of the economic activity, while Kim and Nelson (2001) extended the univariate Markov switching to the multivariate model.
Another class of regime switching models is a Threshold autoregressive (TAR) model. This model was developed by Tong (1978) in time series settings. TAR depends on a few nonlinear elements regularly experienced practically, for example, increasing or decreasing asymmetry designs in a given time series. Ordinarily, the TAR model can be portrayed as a set of various linear AR models, with regime switches happening because of the development of threshold variable(s) with respect to fixed threshold(s). More particularly, in parts, the TAR model utilizes linear models to get a favoured estimation of the conditional mean. In spite of the fact that the TAR model is mostly linear, the likelihood of regime switching suggests a general nonlinear conduct for a given time series (Xaba et al., 2015).
According to Brooks (2001), TAR models have been less widely used than conditional variance models in the arena of economics and finance despite that these models are becoming more popular. The models are clearly of importance when the data may be drawn from one AR model in one regime, but an entirely different autoregressive model in another. Enders (2008) defines a TAR model as a regime switching model that allows the behaviour of a time series say (Xt) to
depend on the state of the system. The TAR model parameters can be estimated using the ordinary least square (OLS) approach. In addition, one side of the threshold (Xt) sequence is governed by one AR process and on the other side there is a different AR process. Although (Xt) is linear in each regime, the likelihood of regime switching means that the entire (Xt) is nonlinear.Jonathan
and Kung-Sik (2008) had also indicated that the value of the threshold is unknown and it must be estimated along with the other parameters of the TAR model.
Another family of regime switching models is the Smooth Transition Autoregressive (STAR) model which was developed by (Chan and Tong, 1986). The STAR model can be thought of in terms of the extension of AR model, allowing for changes in the model parameters according to the value of weakly exogenous transition variable Zt. The model comprises of two AR parts
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depicting the transition function. In particular, p is the order of the AR part. The most well-known transition function incorporates the exponential function and first and second-order logistic functions which in turn offer ascent to logistic STAR (LSTAR) and Exponential STAR (ESTAR) models (Tong, 2012).
2.5 Markov switching model and extensions
This section provides a description and discussion of MS model and the extensions.
2.5.1 AR enhanced Markov-switching model
Because many time series occasionally exhibit dramatic breaks in their behaviour, Hamilton (2010) has indicated that MSM are quite amenable to theoretical calculations of how these abrupt changes in fundamentals show up especially in financial time series. To describe the consequence of a dramatic change in the behaviour of a single series say Yt, Ang and Bekaert (2002) and Dai et al. (2007) have discovered that the behaviour of the past can be described by the first order autoregressive denoted by AR(1). Due to the failure of the AR(1) to account for the change in the parameters, the current study establish a larger model encompassing the change in the parameter and the model must be estimated as:
Yt = Cst+ ΦYt−1+ εt (2.1)
where Cst is the random variable that has happened as a result of institutional changes with values starting from st = 1 for t = 1,2, … , t0 and st = 2 for t=t0+ 1, t0+ 2, …. A full description of the dynamics of Yt could be obtained if a probabilistic description of how the economy changes from
one regime to another is available. According to Hamilton and Raj (2013), the simplest model would be an MC written as:
Pr(St = j|St−1= i, St−2 = k, … . , Yt−1, Yt−2, … ) = Pr(St= j|St−1 = i) = pij. (2.2)
With the assumption that St is directly unobservable, its operation can be inferred through the
observed behaviour of Yt. According to Timmermann (2000), the probability law governing the parameters it fully follows the Gaussian innovation σ2, the coefficient of an autoregressive Φ, the
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two intercepts C1 and C2 and finally the two state transition probabilities p11 and p22 that follow
homogenous MC.
An attractive feature of the model is that no prior information regarding the dates when the economy was in each regime, or the size of the two growth rates is required. This is in contrast with models such as probit and logit models that require and depends heavily upon the exact dates of all the regimes in the history of the series. Instead, the probability of being in a particular regime is inferred from the data (Moolman, 2005).
It should be noted that the way things are, the model needs a confinement to uniquely characterize the states and the state specific parameters. Generally, economic instinct or the purpose of the analysis itself yields such a confinement. Thinking of restrictions over time, a natural identification will be associated with the first state of periods of low inflation rate and second state with periods of high inflation, i.e. ordering the state specific parameters according to some element of Φ such that Φ1i< Φ2i of which i is either a value out of 1, … , R. Kaufmann (2002) has shown that there
is a problem of choosing the parameters as for which the states are recognized. Possibly, the states may even be ineffectively separable in terms of Φ, but well separable in terms of the persistence parameters ηii, i = 1,2, in which ηij is defined in (2.2) and a more generalisation is:
ηij = p(St= j|St−1 = i) (2.3)
Restricting, η such that η11 < η22 this would then help in identifying the state specific parameters. Sometimes, even a combination of the two might be appropriate see (Kaufmann, 2000).
In practice, if the process is not irreducible, all states are visited with non-zero probability in the steady state. This means that the moment analysis can simply be conducted on the subset of states occurring with non-zero stationary probability, i.e an analysis of the unconditional moments starting from the steady state need not assume that the Markov process is irreducible. This is so as either a single state or a block of states in absorbing all other states and therefore it will have zero steady state probabilities (Timmermann, 2000). In this case, MC (2.2) act as if the shift from C1 to C2 is a deterministic event and therefore the permanence of shift would be represented by p22 = 1 though the Markov formulation which invites a more general possibility that p22< 1.
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A model of the form of (2.1)-(2.2) with no AR elements, i.e Φ = 0 was first invented by Baum et al. (1970), while Hamilton (2010) incorporated the AR elements and described such a process as a “Hidden Markov models”(HMM). The formulation of the problem described here, in which all the objects of interest are calculated as a by-product of an iterative algorithm similar to Kalman filter is due to (Hamilton, 1989). In instances where Yt is observed directly, inferences about the St values takes the form of two probabilities such that:
ξjt = Pr(St= j|Ωt; θ), (2.4)
for j=1,2, and the sum of these probabilities is at unity. Ω𝑡 denotes the set of observations obtained at t and θ is a vector of population parameters which the general presentation is θ = (σ, Φ, C1, C2, p11, p22)`. Francq and Zakoıan (2001) indicated that the inference is performed
iteratively for t = 1,2, … , T with step t accepted as input values:
ξi,t−1= Pr(St−1= i|Ωt−1; θ) (2.5) for, i = 1,2 and this produce the same output as in (2.2). The key magnitudes one needs in order to perform this iteration are the densities under the two regimes which are calculated like:
ηjt = f(Yt|St = j, Ωt−1; θ) = 1 √2πσe [−(Yt−Cj−ΦYt−1) 2 2σ2 ] (2.6)
These industriousness parameters are essential in determining the higher order snapshots of the MS process. Moreover, autoregressive parameters are brought into the procedure as in the second and third switching models and extended to vector autoregressive form. This offers ascend to cross-product terms that upgrade the arrangement of third and fourth order moments and the patterns in serial correlation and volatility dynamics that these models can produce.
At the point when the MS models are considered as data producing forms, Yang (2000) demonstrates that it is alluring to clarify the generated MS processes by the conventional stationary-nonstationary basis and to describe their autocovariance structure. Economic theories frequently propose that disparities in economic harmony relationships ought to be stationary after some time. The known possibility for such stationary inconsistencies incorporates stationary
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ARMA and fractional ARMA process. Compared with these stationary procedures, the MS process seem more adaptable as a structural shifts and short-run asymmetric responses can be portrayed efficiently (Hamilton, 2010). In the event of stationarity, the MS procedures will give an adaptable option system to model the disparities in economic equilibrium relationships. Accordingly, finding and describing stationarity conditions for the MS models seem imperative (Yang, 2000).
2.5.2 VAR enhanced Markov-Switching model
The extension of a univariate MSM to multivariate models was pioneered by many scholars which among them include Kim and Nelson (2001) and Droumaguet (2012) the new extended model is known as Markov-Switching Vector autoregressive (MS-VAR) which is the extension of (2.2-2.3) respectively. The observed time series is of a vector Yt = (Y1t, … , YT)′ whose parameters are
unconditionally time varying but constant when conditioned on an unobservable discrete regime variable St ∈ {1, … , M}. A Pth order M-state MS-VAR of K-dimensional time series vector is
denoted by MS − VAR(p). Which therefore yield:
Yt = μSt+ ∑Pi=1ΦSt(i)Yt−1+ εt (2.7) The error term of the MS-VAR follows the Gaussian distribution that is conditioned on St: εt|St~NID(0, ∑(St)). As per Francq and Zakoıan (2001), the parameter shift functions such as
μ(St), Φ1(St), … , Φp(St) and ∑(St) describes the dependence of the vector autoregressive (VAR)
parameters on the regime variable 𝑆𝑡 as:
{ μ1 = (μ11, … , μk1)` if S t = 1 ⋮ μM = (μ1M, … , μkM)` if St= 2 (2.8)
The decisive characteristic of the MS model is that unobservable realisations of the regime such that St∈ {1, … , M} are generated by a discrete time, discrete state and Markov state stochastic process defined by the following transition probabilities:
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As normalization, the regimes are ordered with increasing mean, i.e ‖μ1‖ < ‖μ2‖. Identification
requires that for all m ∈ {1, … , M}, there exist at least one k ∈ {1, … , K} such that μkm ≠ μ1i for i ≠ m. Bayesian methods are incorporated in this study as to compute the value of N implying the largest value for the marginal likelihood and the highest Bayes factor and adopted by Krolzig (2003) who enhanced MS-VAR model and introduced Bayesian priors. In addition, he used the model to explore the regime switches of inflation rates of South Africa. This model is basically used as a precursor to the logistic model for the current study. The intention is to obtain the regime switches and factor them into a logistic regression framework which is later used as a predictor of inflation crisis in South Africa.
The MS-VAR has two conditional assumptions which characterize it. (1) The first order MC where its development is autonomous of the past series is expected to be a series type of {St} with the conditional distribution of latent series given the estimation of {St}t′< t and it just relies upon the
latent series with one lag. (2) The inflation rate is an AR process of order P > 0 with the coefficients advancing in time with the sequence of the inflation type of {Xt}t′ < t and {S
t}t′< t
is the conditional spreading of Yt.
Zhou et al. (2014) constructed MS-VAR model to measure the nonlinear correlation between stock returns in Shanghai, Hong Kong and America, finding differing characteristics in the correlations among markets and various dynamic causal relationships. The MS-VAR approach adopted in this paper then also takes inspiration from the work of Rodriguez (2007), Zhou et al. (2014) and Troug and Murray (2015).
The advancement of Hamilton (1989, 1990) MSM offered ascent exploration of the exchange rate dynamics movements in the studies of currency crises. The modelling of EWS for currencies through the implementation of Markov-Switching Regression (MSR) with two considered regimes together with stable and volatility as a mixture of two normal distributions was first modelled by (Engel and Hakkio, 1996). Nevertheless, time-varying transition probability of the MSR model demonstrated an impermeable of pollution in Asian financial crisis.
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2.6 Empirical literature
This section of the study evaluates the empirical literature as part of the selected models for the study.
2.6.1 Empirical literature on Markov-switching vector autoregressive model
The studies of financial models have gained more practice in recent years. Nonlinear models such as Markov-Switching have twisted out to be increasingly the mainstream of industrial economic and financial studies. Breaking it down into pieces, this includes among them industrial production, business cycle issues, interest rate, unemployment rates and stock prices among others. Because MSM are often adopted by researchers with the aim to account for specific features of economic time series such as the asymmetry of economic activity over the business cycle, Timmermann (2000) has indicated that these features translate into the higher order moments and serial correlation of the data generating process, so a characterization of the moments and autocorrelation function generated by MS will allow researchers to better understand when to make use of this class of models.
On the other hand, Ailliot and Monbet (2012) indicated that the Hamilton’s Markov-Switching is an algorithm of drawing the probabilistic inference of discrete shifts of the mean growth rate of a nonstationary series in the form of a nonlinear interactive filter. The estimation of population parameters by a maximum likelihood method provides the foundation for forecasting future values of the series through the filter permission.
Furthermore, the stochastic instability segment of heterogeneous durations is been joined by the assets returns of the MSM as stated by (Calvet and Fisher, 2004). Lux (2008) has introduced a MS multifractal model for which the model allows the estimation of the parameters through the maximum likelihood estimation (MLE) and Bayesian forecasting of volatility. In any case, the appropriateness of MLE is confined to cases with a discrete distribution of volatility segments. From a handy perspective, MLE, likewise, turns out to be computationally unfeasible for huge quantities of segments regardless of the fact that they are drawn from a discrete distribution, therefore the contribution of the study too advance MLE for parameter estimation to expectation maximum (EM).
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Furthermore, the inflation crises series instability of standard models can be determined using the MSM (Droumaguet, 2012). Studies proved that forecasting volatility with models such as fractionally integrated generalized autoregressive conditional heteroscedasticity (FIGARCH) gives biased results due to its unstableness. Researchers further used MSM for financial industries with the aim to model and forecast volatility of the financial sector and also compute the risk and price derivatives because of the shortcomings of the FIGARCH and other volatility models which cannot perform the task.
Engel and Hakkio (1996) used the MSM model while modelling the European exchange rate volatility. Firstly, the authors extended Hamilton’s model by allowing the probability of switching from one state to another to depend on position of the exchange rate within its European Monetary system (EMS) as opposed to Hamilton’s assumption of constant switching probabilities. This model of Engel and Hakkio was then propagated in the early warning system studies such as of Cruz and Mapa (2013) to identify episodes of high and low inflation in which the outcome of the regime classification appeared to be erratic with the regime lasting for a month.
Abiad (2007) used MS-VAR to identify and characterise the crisis period endogenously. In terms of the Asian countries currency, crises and potential determinants of exiting the tranquil state was tested and a number of variables with significant medians across the panel were found. By using the panelised maximum likelihood methodology, Arias and Erlandsson (2004) in their study of regime switching as an alternative of early warning system of currency crises found that the method allowed them to extract smoother transition probabilities than in the standard case, reflecting the need of policy makers to have advance warning in the medium to long term rather than the short term. See also Brunetti et al. (2008); Abiad (2003) and Mariano et al. (2002).
Two primary issues have been faced by past analysts. Firstly, much research has been developed around the significance of precision in deciding the timing and duration of crisis periods. Furthermore, there has been a significant debate by researchers endeavouring to decide the most ideal approach to analyse correlation dynamics before, during and after these crisis stages (Troug and Murray, 2015). Scholars like Forbes and Rigobon (2002), Boyer et al. (2006) and Rodriguez (2007) had problem in accurately determining the crisis. These authors utilized different methods such as exogenous and endogenous approach but all found different results. Rodriguez (2007)
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found evidence of changing dependence structures during periods of financial turmoil. Increased tail dependence and asymmetry in times of high volatility characterize the Asian countries, while symmetry and tail independence describe better the Latin-American case. Baur (2012) notes that the length of the crisis periods, when determined endogenously, far exceeds those determined using exogenous methods.
In order to reduce the selection bias of the crises, the study adopts Rodriguez (2007) method of using an MS model to endogenously predict an inflation crisis period of several months, which is same with that of (Cruz and Mapa, 2013). The approach utilized as a part of the exploration in the study includes consolidating the MS techniques utilized by Rodriguez (2007) with the VAR methodologies of Forbes and Rigobon (2002) empowering more accurate modelling of the crisis period, in view of lagged values of inflation and repo rate. This MS-VAR approach has beforehand been actualized effectively by Brandt et al. (2012) to determine crisis periods inside datasets for Israel and Palestine.
Moysiadis and Fokianos (2014) noted that a MC algorithm has given the future states and its significance while the categorical response variable is lagged. Two reasons arise for the problem caused by a clear categorical time series when it is modelled by Markovian methods. (1) There is a positive nonlinear relationship between the order of MC and free parameters. i.e., as the order of the MC accumulates, so does the free parameters. Nonetheless, these free parameters increase exponentially. However, the study incorporates the Bayesian approach to restrict these free parameters to increase exponentially but remain constant over time with non-constant regime switching probabilities (2). The response variable and the covariates which are observed jointly must be a Joint flow between them. Of course, this type of determination might be impossible in the stochastic processes of higher time series frequencies.
With the point of depicting and forecasting financial time series volatility from one day to one month skyline, Marcucci (2005) thought about various generalised autoregressive conditional heteroscedasticity models. The unreasonable tirelessness which is typically found in generalised autoregressive conditional heteroscedasticity (GARCH) models are been modelled by Markov-switching GARCH (MS-GARCH) where the parameters are permitted to switch between the low and high volatility regime. Nonetheless, the final results of Marcucci (2005) signposted that
MS-19
GARCH models do truly beat all standard GARCH models in estimating volatility at shorter skylines as indicated by broad set of statistical loss functions but models do really outperform all standard GARCH models in forecasting volatility at shorter horizons. At longer skylines standard deviated GARCH models fare the best.
Sims and Zha (2006) used regime switching within a structural vector autoregressive (SVAR) to assess the impact of changes in the US monetary policy. Their model allowed the time variation in disturbance variances only. With coefficients allowed to change, the model is with change only in the monetary policy rule with estimating the three regimes which are corresponding roughly to periods when most observers believe that monetary policy actually differed. But the differences among regimes are not large enough to account for the rise, then decline, in inflation rate. Droumaguet (2012) extended the model of Hamilton (2008b)
Hamilton (2008a) extended a univariate modelling to multivariate with up to 20 equations. MS-VAR models considered for stage Monte Carlo experiment are the models with switching intercepts. Droumaguet (2012) scrutinized three classes of models which are models with regime switching in intercepts only, models with regime switching in the variance only and model with regime switching in all the parameters that is, switching in intercept vector, autoregressive coefficients matrix and variance-covariance matrix. Ang and Bekaert (2002) showed that incorporating more series in the model provides better regime classification than in the univariate case. This study hence estimates the latent regime in a multivariate case in the current study.
Sims et al. (2008) present the Bayesian methodology for handling general Markov Switching Structural Vector Autoregressive (MS-SVAR) models. These models were found to be useful in business cycles for instance. Thus they are potentially suited in many cases where SVAR models are traditionally used. Kilian (2006) on the other hand introduced Bayesian impulse response analysis for MS-VAR model. The obtained regime separated the sample into two periods over the periods of 1986. The structural changes that occurred in time transformed the oil market into more competitive as highlighted within the regime dynamics.
Krolzig (2000) focused on the predictability of MS-VAR processes as the property of a stochastic process in relation to an information set. They derive the optimal predictor, and show that its properties depend on (i) the significance of regime movements, (ii) the tirelessness of the regime
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generating process, (iii) the asymmetry of the regime generating process and (iv) the interaction with the AR elements. The outcomes obtained permit to infer parametric conditions under which the optimal predictor shrinks to a linear forecast prediction rule.
To investigate the inference and volatility forecasting, Chen et al. (2009) used a Markov-Switching generalized autoregressive heteroscedastic (MS-GARCH) model with a fat-tailed error distribution. This was done to allow the analysis of asymmetric effect on both the conditional mean and conditional volatility for the financial time series. The motivation of extending the model to MS-GARCH model is that the switching variable is said to follow unobserved first-order Markov process and this simultaneously incorporates a MS in variance and mean of the model. Furthermore, these inference and estimation of parameters are executed through the MCMC scheme which follows a Bayesian framework. In addition to that, Chen et al. (2009) used Bayesian forecasting in their comparative study of value-at-risk and all the proposed methods were demonstrated via simulations and eight international stock market return series. The findings of the study indicated that a double MS-GARCH model with an exogenous variable outperformed all other proposed models.
Moolman (2005) applies a Markov regime-switching model to assess the relationship between stock returns and macroeconomic variables in South Africa. The author finds that the degree to which stock returns depend on macroeconomic variables, depends on the state of the business cycle in South Africa. Apart from being used to capture cyclical asymmetry in the stock market, the MSM can also be used to identify turning points in the economy and to model economic growth.
2.6.2 Empirical literature on Logistic regression model
This study attempts to predict the likelihood of inflation crisis in South Africa. In doing so, the logistic regression model is estimated. According to Edison (2000), the model known as an early warning system (EWS) is engaged for the prediction of crises mainly the financial crises. There are various types of crises of which the 2008 United States (US) financial crises typically studied by Kenourgios et al. (2011), including currency crises which were studied by Jeanne and Masson (2000), banking crises by Borio and Drehmann (2009), sovereign debt crises and private sector
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debt crises Schimmelpfennig et al. (2003), and equity market crises by (Bekaert et al., 2011). Therefore, the study extends the current focus of crises to the prediction of inflation crises.
When finding signs and likelihood of classification for the high and low inflation episodes in South Africa, the study proposes the logistic regression. Logistic regression is a model where the dependent variable (DV) is categorical (Freedman, 2009). While fitting the logit model, James et al. (2013) viewed the maximum likelihood estimate (MLE) as an optimal technique in fitting the best logistic model.
In developing an EWS for inflation crisis, there are three methodologies that emphasizes. These are the bottom-up methodology, the aggregate methodology and the macroeconomic methodology. The odds of inflation crisis are addressed and the systemic volatility is being activated and signed if the odds become significant. For the second method, the model is applied to data other than individual banking data. On the third method, the focal point is centered in building a relationship between economic variables with the review that various macroeconomic variables are required to affect the financial system and reflect their own condition.
To describe the conditional log-odds of the conditional variances, the time series that trails a categorical setting is been pushed by an inactive procedure as Moysiadis and Fokianos (2014) emphasised in their study of binary time series modelling. Here, the problem of ergodicity, stationarity and MLE were studied with the estimation of a multinomial logistic models which include latent process.
The application of logistic regression analysis in the prediction of bankruptcy has been pioneered by Ohlson (1980). The logit methodology Joins together the nonlinear changes and makes use of the cumulative distribution function from the logistic to maximize the Joint likelihood of default for the firms’ upset and also, the non-failure likelihood for the sound organizations in a sample. A great part of the early research in the region of financial suffering, concentrated on Multiple Discriminant Analysis (MDA) and after that in later years on logistic analysis (LA).
Davis and Karim (2008) used a multivariate logit model in their comparison study of an early warning system with the aim of relating the likelihood of occurrence or non-occurrence of a crisis to a vector of 𝑛 explanatory variables. The probability that the dummy variable takes a value of
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one (crisis occurs) at a point in time was given by the value of the logistic cumulative distribution evaluated for the data and parameters at that point in time. Their results showed that the logit model they estimated maybe the best model for globally detecting the banking crisis.
Although models have been developed to allow banking crisis prediction, their comparative performance is difficult to evaluate. Current models have been derived from various historic datasets and more importantly, by using different dependent variables and overall methodologies. Consequently, leading indicators may appear inconsistent and in-sample and out-of-sample results differ. The study contribution includes the following: (1) The study makes a follow-up of both Davis and Karim (2008) and Cruz and Mapa (2013), by estimating the logistic regression to predict the inflation crises in South Africa. (2) The cross-country and time-series coverage is more extensive than most previous studies. Hence consideration of refinements to the current EWS by considering how all other classified crisis theory together with the monetary transmission mechanism theory could help to improve specification and variable choice.
Moreover, Gunsel (2005) indicated that the financial ratios and failure of banks in North Cyprus have some sort of relationship which has been linked by a multivariate logit model. The main point of using a multivariate logit model is to compute the probability of bank failure as a vector explanatory variables. Basically, financial ratios are designed to measure information for the six categories which the natural potential risk within the financial institutions is emphasized. In spite of the potentiality to predict the sign of the crisis, binary time series models have been considered for this purpose Further than that, financial indicators are mostly used as bank indicators.
Due to the small samples and the need to keep the degrees of freedom, Kolari et al. (2002) added to the work of EWS by estimating the stepwise logistic regression in order to identify the subset of the covariates that are needed in the model through their power to discriminate. The predefined significance level was set at 10% and the impact of this was that few variables were chosen in the model, hence the need to increase the significance level to 30 which now was used as a threshold to add variables in the model. The main problem that caused the lack of significance of the variables in entering the model is due to the fact that the error term in the regression model followed a cumulative distribution which does not accurately estimates a logit function.
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2.7 Concluding remarks
This chapter has reviewed both theoretical and empirical literature of the Markov-Switching model and Logistic regression model. Both models have been indicated as good models for EWS as Markov-Switching model is identified as a best model for abrupt change modelling in the macroeconomic and financial time series data. Past researchers have been using the MS model for modelling financial returns volatility. However, the current study uses the MS to estimate the periods of high inflation. The MS model is used to cater again for the differences in the equilibrium of inflation rate and repo rate. For model estimations procedure, literature has emphasized that MLE has some deficiency of feasibility while estimating a large series with discrete modelling such as MSM therefore the best approach is through migrating from MLE to EM.
There are two methods of estimating the switching which are identified as constant switching probabilities and varying switching probabilities. The varying switching probabilities allow the researcher to establish better smooth transition probabilities as opposed to the constant switching probabilities. Furthermore, in determining inflation crises in the current study, the variables are set to be endogenous as this is identified as an optimal way of predicting the crises and correctly classifying them.
While building an early warning system, some studies such as of Fuertes and Kalotychou (2007), used K-means clustering while the model for EWS in this study is motivated by the methodology of Cruz and Mapa (2013) who used the univariate logistic regression model to classify the episodes of high inflation accordingly. Again, the literature has highlighted Monte-Carlo experiments plays a vital role in the empirical analysis of econometric modelling when especially the properties of the original data, or models at study are well known and properly predefined.
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CHAPTER 3
METHODOLOGY
3.1 Introduction
Time series analysis contains procedures for analysing time series data keeping in mind the end goal to separate important measurements and different attributes of the data. According to Jonathan and Kung-Sik (2008), the motivation behind time series analysis is by and large twofold: (1) To comprehend or demonstrate the stochastic mechanism that offers ascend to an observed series. (2) To predict or forecast the future values of a series in light of the historical backdrop of that series and perhaps, other related series or variables. The present study delineates diverse methods prepared to help out for the perception of the issue.
Using proper models and doing analysis is the key aspect that is discussed in this chapter. The study integrates two modelling methods for building an early warning system for inflation. An R programing 3.2.4 revised software Matlab and Oxmetrics 6 are used to execute the analysis. The results are presented in tables and figures. The analysis of data is geared to achieve the following objectives which are to:
• Simulate the inflation and repo rates data using Monte Carlo method,
• Estimate a Markov-Switching Bayesian vector autoregressive model using the inflation rate and repo rate of South Africa,
• Estimate the logistic regression model as a classifier of inflation crises on the basis of repo rate,
• Formulate suggestions for policy and future studies.
The rest of this chapter is organised in the following chronological order: Section 3.2 presents preliminary data analysis methods where nonlinear tests, descriptive statistics of the data and normality test are addressed. Section 3.3 presents prior tests and the discussion of information criterions used for maximum lag length. Section 3.4 discusses the proposed methods used to address the study objectives. Section 3.5 presents model diagnostic checking. Section 3.6 discusses
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the forecasting procedure using the selected MS-BVAR model together with the forecasting performance error metrics. And lastly section 3.7 presents the concluding remarks for the chapter.
3.2 Data simulation
The process used for generating the data used in this study follows a nonlinear VAR mechanism of order 1 to 2 (Smith et al., 2013). Here, μt~i. i. d N(0,1) random variables. Again, St is a two-state standardised, irreducible and periodic MC. In these experiment, the study generates 50+T data points for both the inflation and repo rate with the starting values being set to zero and 𝑆0=1
for both the series. In order to attenuate the effect of the initial values, the last T = 1 is in the Monte Carlo replication. It is worth noting that random deviations are a product of the algorithm of (Kinderman and Ramage, 1976).
The sample is chosen based on the three criterions or selection methods. Firstly, the parameters and biases in standard error are less than 10% for any parameter estimated. Secondly the standard error bias for the parameter whose power is being evaluated is less than 5%. In the last step, the coverage should be in the range of 0.91 and 0.98. Provided the three steps are covered, the sample size is chosen to keep power close to 0.80 (Dell et al., 2002). According to the authors, 0.80 is a generally conventional value for sufficient power. Muthén and Muthén (2002) also suggested 0.8 as the best threshold for sample size and model power determination.
The generated sample start at 1 to 𝑛 and following Dendukuri et al. (2004); Mustafa (2006), the chosen sample converges when ε ≥ |x̅ − μ| with confidence interval constructed as 1 − α = pr (|x̅ − μ| < zα
2 ⁄
σ
√n) where 𝑧𝛼⁄2is the upper 100(1 − α2)
th percentile of the standard normal
distribution. Relying on the confidence interval, the deterministic sizes are determined by:
nCLTa = (ε, α, σ2 < ∞) = ⌈(zα 2 ⁄ σ ε) 2 ⌉. (3.1)
Here, ε is the margin error of estimation of the sample.
The application of Monte Carlo methods and simulation has rapidly increased especially in science related studies. These methods are also of ut-most importance in growing sub-disciplines such as those of computational physical sciences and the computational sciences. Recent technological
